R.C.C. FOOTINGS

There are mainly two types of R.C.C. footings:

1. One way reinforced footings.
2. Two way reinforced footings.

1. One Way Reinforced Footing: These footings are for the walls. In these footings main reinforcements are in the transverse direction of wall. In longitudinal directions there will be only nominal reinforcement.

2. Two Way Reinforced Footings: For columns two way reinforced footings are provided.
The following types of the footings are common:

(i) Isolated Column Footings: If separate footings are provided for each column, it is called isolated column footing. Figure 1 shows a typical isolated column footing. The size of footing is based on the area required to distribute the load of the columns safely over the soil . These footings are provided over a 100 to 150 mm bed concrete. Required reinforcements and thickness of footing are found by the design engineers. Thickness may be uniform or varying.

(ii) Combined Footings: Common footings may be provided for two columns. This type of footing is necessary when a column is very close to the boundary of the property and hence there is no scope to project footing much beyond the column face. Figure 2 shows a typical combined footing. The footing is to be designed for transferring loads from both columns safely to the soil. The two columns may or may not be connected by a strap beam.

(iii) Continuous Footings: If a footing is common to more than two columns in a row, it is called continuous footing. This type of footing is necessary, if the columns in a row are closer or if SBC of soil is low. Figure 3 shows this type of footing.

(iv) Mat Footing/Raft Footing: If the load on the column is quite high (Multistorey columns) or when the SBC of soil is low, the sizes of isolated columns may work out to be to such an extent that they overlap each other. In such situation a common footing may be provided to several columns as shown in Fig. 4 Such footings are known as raft footings. If the beams are provided in both directions over the footing slab for connecting columns, the raft foundations may be called as grid foundation also. The added advantage of such footing is, settlement is uniform and hence unnecessary stresses are not produced.

CONVENTIONAL SPREAD FOOTINGS

This type of foundations are commonly used for walls and masonry columns. These foundations are built after opening the trenches to required depth. Such footings are economical up to a maximum depth of 3 m. As these foundations are suitable depth, they are grouped under shallow foundations.

Figure 1 shows a conventional spread footing for a wall and Fig. 2 shows it for a masonry column.

Before building these footing trenches are opened to required depth and the soil is rammed well. Then a plain concrete of mix 1 : 4 : 8 is provided. Its thickness varies from 150 to 200 mm. Over this bed, stone masonry footing is built. It is built in courses each course projecting 50 to 75 mm from the top course and height of each course being 150 to 200 mm. In case of wall footing the projections are only one direction while in case of columns, they are in both directions. The projection of bed concrete from the lowest course of foundation masonry is usually 150 mm.

Design Example: Trench Fill Strip Footing.

The internal load-bearing wall for a four-storey office block is to be supported on a strip foundation. Borehole investigations produced the consistent soil profiles shown in Fig. 11.13.

Soil analysis shows that the sand fill is an unreliable bearing strata. The weathered sandstone has net allowable bearing pressures of na = 400 kN/m2 for strip footings and na = 550 kN/m2 for pads, both with a maximum of 20 mm settlement. The sandstone bedrock has a net allowable pressure of na = 2000 kN/m2
for pad foundations.

By inspection of the soil profile and analysis in Fig. 11.13, the strip will be founded in the compact weathered sandstone. The relatively even distribution of the loading will not lead to unacceptable differential settlements and, as the sides of the excavations do not collapse in the short-term, mass concrete trench fill footings have been selected as the most appropriate foundation type.

Fig. 11.13 Borehole log for Design Examples 1, 2 and 4.

Loadings
The loadings from the four-storey structure have been  calculated (as working loads) as follows.

Size of base (normal method)
The foundation surcharge is considered small enough to be neglected. The minimum foundation width is given by

In many instances this approximate method is satisfactory.

Where the new foundation surcharge is large, or the allowable bearing pressure is low, the following method should be used.

Size of base (allowing for foundation surcharge)
Dead load from new surcharge

Imposed load from new surcharge

The weight of the new foundation is taken as approximately equal to the weight of soil displaced, and thus is
excluded from the above loads.

The net bearing pressure is

In this case the existing surcharge sS = 0.

As may be seen, the normal method value of B = 0.71 m in this example is sufficiently accurate for all practical purposes.

Final selection of foundation width must take into account the width of the wall, together with an allowance for tolerance. It should also try to suit standard widths of excavator buckets which are in multiples of 150 mm, e.g. 450 mm,  600 mm, 750 mm, etc. In this case a width of B = 750 mm would be appropriate, as shown in Fig. 11.14.

Actual net bearing pressure (ignoring foundation surcharge)

The actual net bearing pressure beneath the strip footing may now be calculated, if required.

Fig. 11.14 Trench fill strip footing design example.

Strip Footings - Typical Examples.

Strip footings are commonly used for the foundations to load-bearing walls. They are also used when the pad
foundations for a number of columns in line are so closely spaced that the distance between the pads is approximately equal to the length of the side of the pads. (It is usually more economic and faster to excavate and cast concrete in one long strip, than as a series of closely spaced isolated pads.)

They are also used on weak ground to increase the foundation bearing area, and thus reduce the bearing pressure – the weaker the ground then the wider the strip. When it is necessary to stiffen the strip to resist differential settlement, then tee or inverted tee strip footings can be adopted. Typical examples are shown in Fig. 1.6.

Fig. 1.6 Strip Footings - Typical Examples.

DESIGN OF COMBINED FOOTINGS BY ELASTIC LINE METHOD.

The relationship between deflection, y, at any point on an elastic beam and the corresponding bending moment M may be expressed by the equation

The equations for shear V and reaction q at the same point may be expressed as

where x is the coordinate along the length of the beam.
From the basic assumption of an elastic foundation

where, B = width of footing, k - coefficient of subgrade reaction.
Substituting for q, Eq. (14.12) may be written as

The classical solutions of Eq. (14.13) being of closed form, are not general in their application. Hetenyi (1946) developed equations for a load at any point along a beam. The development of solutions is based on the concept that the beam lies on a bed of elastic springs which is based on Winkler's hypothesis. As per this hypothesis, the reaction at any point on the beam depends only on the deflection at that point.

Methods are also available for solving the beam-problem on an elastic foundation by the method of finite differences (Malter, 1958). The finite element method has been found to be the most efficient of the methods for solving beam-elastic foundation problem. Computer programs are available for solving the problem.

Since all the methods mentioned above are quite involved, they are not dealt with here.

DESIGN OF COMBINED FOOTINGS BY RIGID METHOD (CONVENTIONAL METHOD).

The rigid method of design of combined footings assumes that

1. The footing or mat is infinitely rigid, and therefore, the deflection of the footing or mat does not influence the pressure distribution,

2. The soil pressure is distributed in a straight line or a plane surface such that the centroid of the soil pressure coincides with the line of action of the resultant force of all the loads acting on the foundation.


Design of Combined Footings

Two or more columns in a row joined together by a stiff continuous footing form a combined footing as shown in Fig. 14.3a. The procedure of design for a combined footing is as follows:

1. Determine the total column loads ∑Q = Q1 + Q2 + Q3 + ... and location of the line of action of the resultant ∑Q. If any column is subjected to bending moment, the effect of the moment should be taken into account.
2. Determine the pressure distribution q per lineal length of footing.
3. Determine the width, B, of the footing.
4. Draw the shear diagram along the length of the footing. By definition, the shear at any section along the beam is equal to the summation of all vertical forces to the left or right of the section. For example, the shear at a section immediately to the left of Q1 is equal to the area abed, and immediately to the right of Q1 is equal to (abcd - Q1) as shown in Fig. 14.3a.
5. Draw the moment diagram along the length of the footing. By definition the bending moment at any section is equal to the summation of moment due to all the forces and reaction to the left (or right) of the section. It is also equal to the area under the shear diagram to the left (or right) of the section.
6. Design the footing as a continuous beam to resist the shear and moment.
7. Design the footing for transverse bending in the same manner as for spread footings.

Figure 14.3 Combined or trapezoidal footing design

 

 

It should be noted here that the end column along the property line may be connected to the interior column by a rectangular or trapezoidal footing. In such a case no strap is required and both the columns together will be a combined footing as shown in Fig. 14.3b. It is necessary that the center of area of the footing must coincide with the center of loading for the pressure to remain uniform.

PROPORTIONING OF CANTILEVER FOOTING.

Strap or cantilever footings are designed on the basis of the following assumptions:

1. The strap is infinitely stiff. It serves to transfer the column loads to the soil with equal and
uniform soil pressure under both the footings.
2. The strap is a pure flexural member and does not take soil reaction. To avoid bearing on the
bottom of the strap a few centimeters of the underlying soil may be loosened prior to the
placement of concrete.

A strap footing is used to connect an eccentrically loaded column footing close to the property line to an interior column as shown in Fig. 14.2.

With the above assumptions, the design of a strap footing is a simple procedure. It starts with
a trial value of e, Fig. 14.2. Then the reactions Rl and R2 are computed by the principle of statics.
The tentative footing areas are equal to the reactions R1 and R2 divided by the safe bearing pressure
qs. With tentative footing sizes, the value of e is computed. These steps are repeated until the trial
value of e is identical with the final one. The shears and moments in the strap are determined, and
the straps designed to withstand the shear and moments. The footings are assumed to be subjected
to uniform soil pressure and designed as simple spread footings. Under the assumptions given
above the resultants of the column loads Ql and Q2 would coincide with the center of gravity of the
two footing areas. Theoretically, the bearing pressure would be uniform under both the footings.
However, it is possible that sometimes the full design live load acts upon one of the columns while
the other may be subjected to little live load. In such a case, the full reduction of column load from
Q2 to R2 may not be realized. It seems justified then that in designing the footing under column Q2,
only the dead load or dead load plus reduced live load should be used on column Q1.

The equations for determining the position of the reactions (Fig. 14.2) are

where R1 and R2 = reactions for the column loads Q1 and Q2 respectively, e = distance of R1 from
Q1, LR = distance between R1 and R2.

Figure 14.2 Principles of cantilever or strap footing design

COMBINED FOOTINGS AND MAT FOUNDATIONS.

We Considers the following types of foundations:

1. Cantilever footings
2. Combined footings
3. Mat foundations

When a column is near or right next to a property limit, a square or rectangular footing concentrically loaded under the column would extend into the adjoining property. If the adjoining property is a public side walk or alley, local building codes may permit such footings to project into public property. But when the adjoining property is privately owned, the footings must be constructed within the property. In such cases, there are three alternatives which are illustrated in Fig. 14.1 (a). These are

1. Cantilever footing. A cantilever or strap footing normally comprises two footings connected by a beam called a strap. A strap footing is a special case of a combined footing.
2. Combined footing. A combined footing is a long footing supporting two or more columns in one row.
3. Mat or raft foundations. A mat or raft foundation is a large footing, usually supporting several columns in two or more rows.

The choice between these types depends primarily upon the relative cost. In the majority of cases, mat foundations are normally used where the soil has low bearing capacity and where the total area occupied by an individual footing is not less than 50 per cent of the loaded area of the building.

When the distances between the columns and the loads carried by each column are not equal, there will be eccentric loading. The effect of eccentricity is to increase the base pressure on the side of eccentricity and decrease it on the opposite side. The effect of eccentricity on the base pressure of rigid footings is also considered here.

SAFE BEARING PRESSURE FROM EMPIRICAL EQUATIONS BASED ON CPT VALUES FOR FOOTINGS ON COHESIONLESS SOIL.

The static cone penetration test in which a standard cone of 10 cm2 sectional area is pushed into the soil without the necessity of boring provides a much more accurate and detailed variation in the soil as discussed in Chapter 9. Meyerhof (1956) suggested a set of empirical equations based on the

Terzaghi and Peck curves (1948). As these equations were also found to be conservative, modified forms with an increase of 50 percent over the original values are given below.

An approximate formula for all widths

where qc is the cone point resistance in kg/cm2 and qs in kPa.

The above equations have been developed for a settlement of 25 mm.

Meyerhof (1956) developed his equations based on the relationship qc = 4Ncor kg/cm2  for penetration resistance in sand where Ncor is the corrected SPT value.

EMPIRICAL EQUATIONS BASED ON SPT VALUES FOR FOOTINGS ON COHESIONLESS SOILS.

Footings on granular soils are sometimes proportioned using empirical relationships. Teng (1969)
proposed an equation for a settlement of 25 mm based on the curves developed by Terzaghi and
Peck (1948). The modified form of the equation is

Meyerhof (1956) proposed the following equations which are slightly different from that of Teng



Experimental results indicate that the equations presented by Teng and Meyerhof are too conservative.

Bowles ( 1 996) proposes an approximate increase of 50 percent over that of Meyerhof which can also be applied to Teng's equations. Modified equations of Teng and Meyerhof are,

Teng's equation (modified),

If the tolerable settlement is greater than 25 mm, the safe bearing pressure computed by the above equations can be increased linearly as,

where q's = net safe bearing pressure for a settlement S'mm, qs = net safe bearing pressure for a settlement of 25 mm.

DESIGN CHARTS FROM SPT VALUES FOR FOOTINGS ON SAND.

The methods suggested by Terzaghi et al., (1996) for estimating settlements and bearing pressures of footings founded on sand from SPT values are based on the findings of Burland and Burbidge (1985). The SPT values used are corrected to a standard energy ratio. The usual symbol Ncor is used in all the cases as the corrected value.

Formulas for Settlement Calculations.
The following formulas were developed for computing settlements for square footings.

For normally consolidated soils and gravels

If the footing is established at a depth below the ground surface, the removal of the soil above the base level makes the sand below the base preconsolidated by excavation. Recompression is assumed for bearing pressures up to preconstruction effective vertical stress q'o at the base of the foundation. Thus, for sands normally consolidated with respect to the original ground surface and for values of qs greater than q'o, we have,

 Figure 13.4 Thickness of granular soil beneath foundation contributing to
settlement, interpreted from settlement profiles (after Burland and Burbidge 1985)

It may be noted here that the ground water table at the site may lie above or within the depth of influence Zl Burland and Burbidge (1985) recommend no correction for the settlement calculation even if the water table lies within the depth of influence Zl. On the other hand, if for any reason, the water table were to rise into or above the zone of influence Zl after the penetration tests were conducted, the actual settlement could be as much as twice the value predicted without taking the water table into account.


Chart for Estimating Allowable Soil Pressure
Fig. 13.5 gives a chart for estimating allowable bearing pressure qs (on settlement consideration)
corresponding to a settlement of 16 mm for different values of TV (corrected). From Eq. (13.6), an
expression for q may be written as (for normally consolidated sand)

Figure 13.5 Chart for estimating allowable soil pressure for footing on sand on the
basis of results of standard penetration test. (Terzaghi, et al., 1996)

The chart m Fig. 13.5 gives the relationships between B and Q. The value of qs may be obtained from Q for any given width B. The Q to be used must conform to Eqs (13.12), (13.13) and (13.14).

The chart is constructed for square footings of width B. For rectangular footings, the value of qs should be reduced in accordance with Eq. (13.10). The bearing pressures determined by this procedure correspond to a maximum settlement of 25 mm at the end of construction.

It may be noted here that the design chart (Fig. 13.5b) has been developed by taking the SPT values corrected for 60 percent of standard energy ratio.

Effect of Size of Footings on Settlement.

Figure 13.3a gives typical load-settlement relationships for footings of different widths on the surface of a homogeneous sand deposit. It can be seen that the ultimate bearing capacities of the footings per unit area increase with the increase in the widths of the footings. However, for a given settlement S, such as 25 mm, the soil pressure is greater for a footing of intermediate width Bb than for a large footing with BC. The pressures corresponding to the three widths intermediate, large and narrow, are indicated by points b, c and a respectively.
The same data is used to plot Fig. 13.3b which shows the pressure per unit area corresponding to a given settlement S1, as a function of the width of the footing. The soil pressure for settlement Sl increases for increasing width of the footing, if the footings are relatively small, reaches a maximum at an intermediate width, and then decreases gradually with increasing width.

Although the relation shown in Fig. 13.3b is generally valid for the behavior of footings on sand, it is influenced by several factors including the relative density of sand, the depth at which the foundation is established, and the position of the water table. Furthermore, the shape of the curve suggests that for narrow footings small variations in the actual pressure, Fig. 13.3a, may lead to large variation in settlement and in some instances to settlements so large that the movement would be considered a bearing capacity failure. On the other hand, a small change in pressure on a wide footing has little influence on settlements as small as S1 , and besides, the value of ql corresponding to S1 is far below that which produces a bearing capacity failure of the wide footing.

For all practical purposes, the actual curve given in Fig. 13.3b can be replaced by an equivalent curve omn where om is the inclined part and mn the horizontal part. The horizontal portion of the curve indicates that the soil pressure corresponding to a settlement S1 is independent of the size of the footing. The inclined portion om indicates the pressure increasing with width for the same given settlement S1 up to the point m on the curve which is the limit for a bearing capacity failure. This means that the footings up to size Bl in Fig. 13. 3b should be checked for bearing capacity failure also while selecting a safe bearing pressure by settlement consideration.

The position of the broken lines omn differs for different sand densities or in other words for different SPT N values. The soil pressure that produces a given settlement Sl on loose sand is obviously smaller than the soil pressure that produces the same settlement on a dense sand. Since N- value increases with density of sand, qs therefore increases with an increase in the value of N.

Figure 13.3 Load-settlement curves for footings of different sizes (Peck et al., 1974)

Ultimate Bearing Capacity for Local Shear Failure: Equations for the Lower Bound Values for the Various Types of Footings.

When a soil fails by local shear, the actual shear parameters c and Ø are to be reduced as per Terzaghi
(1943). The lower limiting values of c and Ø are

The equations for the lower bound values for the various types of footings are as given below.
Strip Foundation

Square Foundation

Circular Foundation

Rectangular Foundation

where Nc , Nq and Nγ are the reduced bearing capacity factors for local shear failure. These factors may be obtained either from Table 12.1 or Fig. 12.7 by making use of the friction angle Ø.

Ultimate Bearing Capacity of Soil: Strip Footings.

Strip Footings
Terzaghi developed his bearing capacity equation for strip footings by analyzing the forces acting on the wedge abc in Fig. 12.6. The equation for the ultimate bearing capacity qu is

where Qult = ultimate load per unit length of footing, c = unit cohesion, γ the effective unit weight  soil, B = width of footing, D,= depth of foundation, Nc, Nq and Nγ are the bearing capacity factor

They are functions of the angle of friction, Ø.

The bearing capacity factors are expressed by the following equations

where Kpy = passive earth pressure coefficient

Table 12.1 gives the values of Nc, Nq and Nγ for various values of Ø and Fig. 12.7 gives the same in a graphical form.

Table 12.1 Bearing capacity factors of Terzaghi

Figure 12.7 Terzaghi's bearing capacity factors for general shear failure

Mechanism of Failure: Terzaghi for a Strip Footing.

The shapes of the failure surfaces under ultimate loading conditions are given in Fig. 12.6. The zones of plastic equilibrium represented in this figure by the area gedcfmay be subdivided into

1 . Zone I of elastic equilibrium
2. Zones II of radial shear state
3. Zones III of Rankine passive state

When load qu per unit area acting on the base of the footing of width B with a rough base is transmitted into the soil, the tendency of the soil located within zone I is to spread but this is counteracted by friction and adhesion between the soil and the base of the footing. Due to the existence of this resistance against lateral spreading, the soil located immediately beneath the base remains permanently in a state of elastic equilibrium, and the soil located within this central Zone I behaves as if it were a part of the footing and sinks with the footing under the superimposed load.

The depth of this wedge shaped body of soil abc remains practically unchanged, yet the footing sinks. This process is only conceivable if the soil located just below point c moves vertically downwards. This type of movement requires that the surface of sliding cd (Fig. 12.6) through point c should start from a vertical tangent. The boundary be of the zone of radial shear bed (Zone II) is also the surface of sliding. As per the theory of plasticity, the potential surfaces of sliding in an ideal plastic material intersect each other in every point of the zone of plastic equilibrium at an angle (90° - Ø). Therefore the boundary be must rise at an angle Ø to the horizontal provided the friction and adhesion between the soil and the base of the footing suffice to prevent a sliding motion at the base.

Figure 12.6 General shear failure surface as assumed by Terzaghi for a strip footing

The sinking of Zone I creates two zones of plastic equilibrium, II and III, on either side of the footing. Zone II is the radial shear zone whose remote boundaries bd and af meet the horizontal surface at angles (45° - Ø/2), whereas Zone III is a passive Rankine zone. The boundaries de and fg of these zones are straight lines and they meet the surface at angles of (45° - Ø/2). The curved parts cd and cf in Zone II are parts of logarithmic spirals whose centers are located at b and a respectively.

⇒ Ultimate Bearing Capacity of Soil 
⇒ Equations for Square, Circular, and Rectangular Foundations
⇒ Ultimate Bearing Capacity for Local Shear Failure
⇒ Ultimate Bearing Capacity qu in Purely Cohesionless and Cohesive Soils Under General Shear Failure

ADJACENT FOOTINGS.

Normally, adjacent footings should be placed at the same level. However, when adjacent footings are to be constructed at different levels, the distance between the edges of the footings shall be such as to prevent undesirable overlapping of stresses in soil and disturbance of the soil under the higher footing due to the excavation of the lower footing. The difficuty can be avoided by keeping the difference in footing elevations
(a) not greater than one half the clear distance (b) between the footings. However, when footing are founded on rock, b should not excedd a.

FIG. 3.36 ADJACENT FOOTINGS AT DIFFERENT LEVELS.

In clayey soils, the line (AB) drawn between the lower adjacent edge of the super upper footing and upper adjacent edge of the lower footing should not have a steeper slope than n1 (horiontal) : 1 (vertical), where n1 is equal to 2.

In granular soils, the line (AB1) drawn between the lower adjacent edges of adjacent footings should not have steeper slope than n2 (horizontal) : 1 (vertical), where n2 is equal to 2.

FOOTINGS AT DIFFERENT LEVELS: STEPPED FOOTINGS.

When the existing ground is sloping and a wall is to be founded over it, it becomes highly uneconomical to provide the base of the footing at the same level all along the length of the wall. In such a circumstance, stepped foundation, such as the one shown in Fig. 3.34 may be provided. The foundation trench is excavated in steps.

The heighta of steps should preferably be not more than the depth of the concrete block and each step should be a multiple of the thickness of brick or stone course. The ovelap between tow layers of foundation concrete should be less than the vertical thickness of concrete.

Another problem of footing at two different levels is illustrated in Fig. 3.35 where a wal footing at the ground
floot adjoint a basement wall. It is common practice to lower the ground floor footings in gradual steps, down to the basement footing as shown.

By doing so, the naturai state of the subsoil is considered unaltered.

FIG. 3.34 STEPPED FOOTING ON SLOPPING GROUND.

FIG. 3.35 WALL FOOTINGS AT DIFFERENT LEVELS.